Discovering Congruent Triangles Activity Objective: Understanding congruent triangle postulates and theorems using inductive reasoning. Materials needed: noodles, protractor, ruler, and construction paper or cardstock Groups: In a group of 2 or 3 Cut noodles into the following lengths: 2 noodles 8 centimeters in length 2 noodles 11 centimeters in length 2 noodles 5 centimeters in length Part 1 1. Put the 3 noodles of different lengths together to form a triangle as shown. 2. Form another triangle with the other set of noodles. 3. Measure the angles of both triangles using a protractor. Questions: 1. What are the measures of the 3 angles in the first triangle? 2. What are the measures of the 3 angles in the second triangle? 3. What is the relationship between the angles of each triangle? 4. Are the triangles congruent? 5. Can the noodles be rearranged to form a triangle with different angles?
Part 2 1. Take 2 of the noodles, place them on a piece of paper, and form a 60 degree angle between them. 2. Take the 2 noodles of the same length and also form a 60 degree angle between them. 3. Draw a line to represent the 3 rd side. Repeat the process for the 2 nd triangle. 4. Measure the length of the 3 rd side and the two remaining angles for each triangle.
Questions: 1. What is the length of the 3 rd side? 2. What are the measures of the remaining angles? 3. Are the two triangles congruent? 4. Use any two noodles and any angle of your choice. Do you get the same result? Will you always get the same result? Part 3 1. Measure three angles measuring 80, 60, and 40 degrees on the corners of 2 pieces of construction paper or cardstock, cut them out, and label them. 80 2. On a piece of paper, take one of the noodles, and place two of the cut-out angles on each end as shown. Repeat the process for the 2 nd triangle. 3. Using a ruler, draw a segment along each of the angle. The two segments should intersect forming the last angle. Repeat the process for the 2 nd triangle.
4. Measure the 3 rd angle and the lengths of the 2 sides in each triangle. Questions: 1. What is the measure of the 3 rd angle for each triangle? 2. What are the measures of the remaining 2 sides for each triangle? 3. Are the triangles congruent? 4. What if you used the 5cm noodle? The 8cm noodle? A noodle with a different length? Part 4 1. Use two of the angles used in the example above. 2. Use one of the noodles and place one of the angles alongside it as shown. Draw a long segment like the dashed one in the drawing. Repeat the process for the 2 nd triangle. 3. Place the second angle along this segment so that when a 2 nd segment is drawn, it will connect with the end of the noodle.
80 80 4. Measure the 3 rd angle and the two remaining sides. Questions: 1. What is the measure of the 3 rd angle for each triangle? 2. What are the measures of the remaining 2 sides for each triangle? 3. Are the triangles congruent? Part 5 1. Place two of the noodles together forming an angle of any degree for one triangle, and repeat the process for the 2 nd triangle. 2. Use one of the pre-cut angles and place alongside the longer of the sides but not as the included angle. 3. Draw a segment to connect the 3 rd side to the other two sides.
4. Swing the 8cm noodle so that it hits the 3 rd side at a different spot in the 2 nd triangle as in the first. 5. Measure the 3 rd side and the remaining 2 angles in each triangle. Questions 1. What is the measure of the 3 rd side for each triangle? 2. What are the measures of the remaining 2 angles for each triangle? 3. Are the two triangles congruent? 4. Do you think that you would get different results if you used a different angle? Part 6 1. Place the 3 angles so that they can form a triangle without measuring the sides initially. Draw segments connecting the angles. Repeat the process for the second triangle. 80 80 2. Measure the 3 sides for each triangle. Questions
1. What are the measures of the 3 sides for each triangle? 2. Are the two triangles congruent? 4. Draw two triangles for each part, and using the correct marks, show which sides and angles are congruent. Match the correct shortcut for each set of triangles from the following choices, and tell whether or not the shortcut is valid for proving triangles congruent. SSS, SAS, AAS, AAA, ASA, SSA S means that the corresponding sides of the triangles are congruent. A means that the corresponding angles of the triangles are congruent.